Abstract
Throughout Volume 2 the electronic properties of solids were studied in the independent electron approximation. The spectrum of Bloch electrons was calculated in the presence of a periodic one-particle potential V(r),1 which could incorporate the contribution of electron–electron interaction at a mean-field level, and the states of the many-body system were obtained by filling the electronic bands successively with independent particles. The interactions with impurities and lattice vibrations could be given a simple interpretation in this picture and we were able to understand some (e.g., transport and optical) properties of insulators, metals, and semiconductors.
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Notes
- 1.
In Volume 2 the notation \(U(\boldsymbol{r})\) was used for the potential. In this volume \(V(\boldsymbol{r})\) stands for the one-particle potential and the notation \(U(\boldsymbol{r})\) is reserved for the two-particle interaction. The subscript e–e will be dropped from \(U_{\textrm{e-e}}\).
- 2.
Contrary to the convention used in semiconductors, in our subsequent treatment of solids bands occupied by electrons that participate in the formation of metallic or covalent bonds will be called valence bands.
- 3.
This model is often referred to as the jellium model.
- 4.
J. Hubbard, 1963. The model is named after Hubbard although the same model was proposed by M. C. Gutzwiller Gutzwiller, M. C. and J. Kanamori Kanamori, J. at the same time to describe the magnetic properties of transition metals.
- 5.
Note that also three-site terms of the form \(c^{\dagger}_{i,\sigma}c^{\dagger}_{j,-\sigma} c^{\phantom\dagger}_{j,\sigma} c^{\phantom\dagger}_{k,-\sigma}\) and \(c^{\dagger}_{i,\sigma}c^{\dagger}_{j,-\sigma} c^{\phantom\dagger}_{j,-\sigma} c^{\phantom\dagger}_{k,\sigma}\) appear in the effective Hamiltonian as the doubly occupied sites are eliminated.
- 6.
The Hartree term is also called direct term.
- 7.
This approximation may give incorrect result in quantum chemistry calculations for atoms or molecules, where the number of electrons is not large.
- 8.
T. Koopmans, 1934. Tjalling charles koopmans (1910–1985) was awarded the 1975 Nobel Memorial Prize in economic sciences.
- 9.
Hartree energy (\(1\,E_{\textrm{h}} = \tilde{e}^2/a_0\)) is the atomic unit of energy. It is conveniently used in the atomic system of units, where the charge and mass of electron, ℏ, and the Bohr radius are taken to be unity, and hence the Hartree energy is also unity. 1 Ry is the binding energy of the electron in the ground state of the hydrogen atom, it is half of the hartree energy.
- 10.
The Wigner–Seitz radius, the radius of the sphere belonging to an electron, was introduced in Chapter 16. Its relationship to \(k_{\textrm{F}}\) is given in (16.2.31).
- 11.
The pair distribution function itself is often called pair correlation function in the literature.
- 12.
j. c. Kimball, 1973.
- 13.
Quite often in the literature, the dynamical structure factor is defined without the factor \(2 \pi /N_{\textrm{e}}\).
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Sólyom, J. (2010). Electron–Electron Interaction and Correlations. In: Fundamentals of the Physics of Solids. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04518-9_1
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DOI: https://doi.org/10.1007/978-3-642-04518-9_1
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